As shown in the four forthcoming examples, both null and time-like geodesics are complete (at least) in the region \({F\ge 0}\) . The non-Riemannian points of \({F=0}\)  form a sphere for \({D>2}\)  (or hyperbola for \({D=2}\) ) inside of which (\({F<0}\) ), no time-like nor null geodesics can penetrate. Time-like and non-radial null geodesics cannot even come close to the sphere from outside (\({F>0}\) ). Only radial null ones may reach the sphere with identically vanishing proper velocities. Besides, undoubled strings, as an alternative probe of the “singular" geometries [1]}, [2]}, become (anti-)chiral: with light-cone coordinates \(y^{\pm }=y\pm t\)  and also \(\partial _{\pm \!}=\!\frac{\partial ~}{\partial \sigma ^{\pm }}\)  on the worldsheet, we get \(\partial _{-}y^{+}=0=\partial _{+}y^{-}\)  (or \(\partial _{+}y^{+}=0=\partial _{-}y^{-}\) ) in the limit \({F\rightarrow 0}\) . Thus, the undoubled, or conventional, particle geodesics and string propagations agree with the non-Riemannian (freezing/chiral/anti-chiral) behaviors predicted from the previous doubled-yet-gauged sigma model approach [3]}, [4]} which relied on the auxiliary gauge connection as Lagrange multiplier.
